Why do my lamps require an insane "strength" value of 1 million?
Your area lamp is not 1m across, it is 100m in size, hence the strength required.

Your Blend file seems to have pretty messed up scaling factors, for starters your viewport has a clipping distance of 50µm which is causing all sorts of display precision issues and Z-Fighting.

Your whole scene is huge in reality, most objects are several meters high, have very high scaling factors, and your room is more than one kilometer long.

Most problems probably stem from the fact that your whole scene has a Units Scale factor of 0.01, which is giving you a false sense of scale, and falsely reported units value

Go to the Properties Window > Scene > Units > Units Scale and set the value to 1